I927
IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. A N D SIGNAL PROCESSING. VOL. 3 8 , NO. 11. NOVEMBER 1990
Efficient Reconstruction of Band-Limited Sequences
from Nonuniformly Decimated Versions by Use of
Polyphase Filter Banks
IEEE, AND
Abstract-Based on recent results on the reconstruction of a bandlimited sequence from a nonuniformly decimated version, this paper
develops an efficient polyphase structure for such reconstruction. Theoretically, the reconstruction involves the implementation of a bank of
multilevel filters, and this paper shows how all these reconstruction filters can be obtained at the cost of one Mth band low-pass filter and a
constant matrix multiplier. The resulting structure is therefore more
general than the ad hoc scheme reported recently. In addition, the
method offers a direct means of controlling the overall reconstruction
distortion T ( z ) by appropriate design of a low-pass prototype filter
P ( z ) . Extension of these results to multiband band-limited signals and
to the case of nonconsecutive nonuniform subsampling are also summarized, along with generalizations to the multidimensional case. Design examples are included to demonstrate the theory, and the complexity of the new method is seen to be much lower than earlier ones.
I. INTRODUCTION
HE theory of nonuniform sampling, and techniques
for reconstruction of a signal from unevenly spaced
samples have a long history dating back to the fundamental paper by Shannon [ 11. Subsequent papers by Papoulis
[2] and Brown [3] throw more light on the topic from a
linear-system viewpoint. An excellent review on this topic
is given in [4].
In a recent article [5, sec. IV], the authors of this paper
have interpreted the reconstruction of a band-limited sequence from a nonuniformly decimated version, in terms
of maximally decimated jilter-bank structures [6]-[ lo].
Before outlining the purpose of the present paper, it is
necessary to first summarize the main result of this interpretation. Suppose x ( n ) is a low-pass sequence which is
band limited to the region 0 I I w I < L a / M where L
and M are integers with L < M . It is then possible to
perform a nonuniform decimation of x( n ) without losing
information, as follows: divide the time axis n into segments of length M , and retain only L consecutive samples
in each segment. Thus, the retained samples are x( n M k ) , 0 Ik 5 L - 1. The signal x( n ) can be reconstructed
from these samples by using a set of L linear time-in-
T
Manuscript received November 7, 1988; revised December I , 1989.
This work was supported in part by National Science Foundation grants
DCI 8552579 and MIP 8604456.
P. P. Vaidyanathan is with the Department of Electrical Engineering,
California Institute of Technology, Pasadena. CA 91 125.
V. C. Liu was with the Department of Electrical Engineering, California Institute of Technology, Pasadena. CA 91 125. He is now with the
Videocipher Division of the General Instruments Corporation. San Diego.
CA 92121.
IEEE Log Number 9038419.
VINCENT
c.
4
LIU,
.
Nonuniform decimator
MEMBER, IEEE
Synthesis bank
Fig. I . The nonuniform decimation and reconstruction scheme
variant filters Fk ( z ) , 0 Ik IL - 1 (collectively called
a synthesis Jilter bunk { FL ( 2 ) } ). The nonuniform decimation and reconstruction can be represented as in Fig. 1
where the symbols 1M and t M denote M-fold decimators
and interpolators, respectively, as defined in [6]-[lo]. In
[5], a closed form expression for the synthesis filters was
derived for arbitrary L , M . The implementation aspects of
the synthesis bank { FL ( z ) } were not addressed, even
though design examples for the special cases of L = 2, M
= 3 and L = 3, M = 4 were presented. The synthesis
filer-bank structures for these special cases were obtained
in a somewhat ad hoc manner, without noticing or exploiting the special properties of these filters to reduce
complexity. Moreover, even though the polyphase approach [ 111 has been used to advantage in other sections
of [5] for other reasons, it was not used in [5, sec. IV],
because the advantage in this specific context was not evident at that time.
A formal mathematical solution for the filters { FL ( z ) }
[ 5 , eq. (56)] reveals that each z-'FL ( z ) is a multilevel
( e J w ) is a (complex) piecewise confilter, i.e., e -JwhFk
stant function of w (more precise statement will be given
in Section 11). We shall use this in Section I1 of this paper
to show that all the L filters { Fk ( z ) } can be implemented
by 1) implementing a single low-pass prototype filter in
terms of its M polyphase components, and 2) forming linear combinations of the outputs of these polyphase components. As a result, the reconstruction cost is equal to
the cost of a single low-pass filter (plus the overhead of
implementing the constant linear combiner, which, as we
shall see, is typically only a small fraction). In Section 111
we consider the L = 2, M = 3 case in order to demonstrate the theory. We include a detailed design example,
and include complexity comparisons with [ 5 ] . More wellknown techniques for fractional decimation and reconstruction (such as the ones in [ 6 , ch. 21) are also included
0096-3518/90/1100-1927$01.OO 0 1990 IEEE
IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. A N D SIGNAL PROCESSING. VOL. 38. NO. I I . NOVEMBER I990
1928
in this comparison. The results of this paper easily extend
to the case when the L samples retained per segment are
nonconsecutive, and also to the case when the signal x ( n )
is multiband rather than low-pass, with total spectral occupancy of 2 a L / M . These extensions, and extensions to
the multidimensional case are summarized in Sections IV
and V.
where [lo]
1 L-l
z-'W"FL(z),
A,(z) = M k = ~
0 5 I
IM -
1.
(3)
If the filters { Fk ( z ) } are chosen to cancel the aliasing
terms (i.e., force A l ( z ) = 0, 1 > 0 ) , then we have
X ( z ) = T ( z ) X ( z ) . Here
A. Notations
Throughout the paper the integers L and M have the
1 L-l
T ( z ) 6& ( z ) = - C z - " F ~ ( z )
(4)
significance described above. Boldface type denotes maM k=O
trices and vectors. The row and column indices begin at
zero. Matrices are often indicated in terms of their entries is called the distortion function of the system. The set of
as A = [ A k m ]where k is the row index and m is the col- synthesis filters which results in perfect reconstruction
umn index. The quantities A*, AT, and At denote, respec- (i.e., i ( n ) = x ( n ) ) is given by the following theorem
tively, the conjugate, transpose, and conjugate transpose 151.
7heorem I : With reference to Fig. 1 let
of A . We denote WM = e-i2n/M, and if the subscript is
omitted it is taken to be M . Any transfer function P ( z )
can be written in the form [ l l ] P ( z ) = Cy=j' z-'PI ( z M )
which is called the type 1 polyphase decomposition. PI ( z )
are unique for fixed P ( z ) and M , and are called the type
(5)
1 polyphase components.
Then
the
set
of
filters
{
Fk
(
z
)
}
which
results
in
perfect
I ) Special Matrices: The M x M DFT matrix is W
recovery (i.e., i ( n ) = x ( n ) ) for all inputs x ( n ) band
= [ Wktn].The L X L leading principal submatrix of W is
denoted as V, i.e., the ( k , m ) entry of V is given by Vk, limited to I w 1 < L a / M satisfies
- Wk'",0 Ik , m IL - 1 . Note that Vis Vandermonde
w E Region i , 0 Ii 5 L - 1
Diui
with distinct columns so that it is nonsingular. We define f ( e J " ) =
otherwise
(6)
U = V-I with the columns of U denoted as U;,so that U
= [ u o uI . . * uL- I ] . The M X L upper left submatrix of
where Region i corresponds to - L n / M + 2 a i / M < w
W is denoted R , i.e., R,,,, = Wktn, O Ik IM - 1 , 0 I < - L a / M
27r(i + l ) / M .
rn IL - 1. We define J to be the permutation of the
The L regions are shown in Fig. 2 for clarity. From (6)
identity matrix with 1's on the cross diagonal. A 3 x 3
we conclude that the filter z - k Fk ( z ) has a constant freexample is
quency response over the Region i. These constant levels
Pki are typically complex, and depend upon both the filter
number k and the region number i. Filters with responses
J = 0 O1 0
of this form are called multilevel filters. They are evidently ideal filters' requiring infinite order (we shall approximate these with practical filters later). We shall show
The ith unit vector [ 0 * * * 010 * * 0 I T with "1" in the in Section 11-A that any such filter can be realized as a
ith position is denoted as e,. Finally, the diagonal matri- linear combination
bl z -'PI ( z M )where PI ( z ) are
ces Dj and A with dimensions L x L and M x M are the type 1 polyphase components of an ideal low-pass fildefined as
ter with response as in Fig. 3(a), i.e.,
0; = diag [ 1, W', . . , W(L-l)' 1
+
lo'1
A
=
diag [ I ,
W(L-l)/2
9
...
I.
W(h'-l)(L-l)/2
(1)
11. POLYPHASE
IMPLEMENTATION
OF THE
RECONSTRUCTION
FILTERBANK{ Fk ( z ) 3
Consider Fig. 1 where x ( n ) is a low-pass signal band
limited to 0 5 1 w 1 < L n / M . Assume without loss of
generality that L and M are relatively prime. The set of L
branches labeled as "nonuniform decimator" retains the
samples x ( n M - k ) , 0 Ik IL - 1. The decimators
and interpolators create aliasing and _imaging effects [6],
respectively. As a result, the signal X ( z ) is a linear combination of X ( z ) as well as the alias components
X ( Z W - ' ) , i.e.,
M- I
X(z)
=
c A,(z)X(zW-')
/=0
So the L filters { Fk ( z ) } can be realized at the cost of one
low-pass filter (plus the cost of linear combinations which
is ML - L2 real multipliers (Section 11-B)).
A. Polyphase Structure for a Multilevel Filter
Let H ( z ) denote the transfer function of a multilevel
filter with frequency response as in Fig. 4 (where 0;s can
'For ideal filters. even though the response can be defined graphically
and FA ( e '") written down, the 2 transform may not converge in the usual
sense. We continue to use the notation FA (:) for convenience, with the
understanding that :is an abbreviation for e
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V A I D Y A N A T H A N A N D LIU: EFFICIENT RECONSTRUCTION OF BAND-LIMITED SEQUENCES
2 KIM
A'
-&\
_$__'
-E
Region L- 1
..
'
I
, i ,
c r r n
*.
t
I
keg1dn 0
W
Fig. 5 . Implementation of the multilevel filter H ( z ) . Here b is the level
vector.
M
Fig. 2. The L regions in
-T 5 w
<
T.
0
M
M
M
(b)
(a)
Fig. 3 . (a) The ideal low-pass prototype and (b) ideal bandpass prototype
.-
2XIM
.. .
PL-1
--pL
...
Note that the quantities GI ( z ) have real coefficients since
the ideal filter P ( z ) (and hence P I ( z ) ) have real coefficients. In practice, one replaces P ( z ) with an approximation, and using (8) one obtains an approximation of the
multilevel filter. The ''levels" are adjustable by changing
the multipliers pi (components of b in Fig. 5 ) without
redesigning GI (z).
Proof of Theorem 2: From Fig. 3 it is clear that P ( z )
= C Lk =- OI B ( z W k ) .By writing B ( z ) in polyphase form i.e.,
as
M- I
B(z) =
Fig. 4. A multilevel response. Here 0, can in general be complex.
be complex). The vector b !
i[ Po PI * . O M will be
called the level vector for the filter H ( z ) . It is clear that
we can generate such a response by repeated use of shifted
versions of an ideal bandpass response B ( e'") shown in
Fig. 3(b). We shall, however, find it more convenient to
obtain H ( z ) starting from the prototype low-pass response P ( e ' " ) , for reasons mentioned in paragraph following Theorem 4; our immediate aim here is to prove
the following result which relates the low-pass response
P ( e'") to the multilevel response H ( e'").
Theorem 2: Let the low-pass prototype P ( z ) be expressed in type 1 polyphase form, P ( z ) = C E i I
z - / P I (zM). Then the multilevel filter H ( z ) with levels PI
as shown in Fig. 4 can be obtained as
=
c
/ = o ?q/z - ' P , ( z M )
P/ ( 4 = 4 & ( z ) .
Similarly, we can express the multilevel filter H ( z ) as
= E?=-, P k B ( z W k )which simplifies to H ( z ) =
EM- I
I = o a l z - ' B I ( z M ) This
.
reduces to (8) in view of (14).
rn
B. Implementation of the Multilevel Synthesis Bank
( F , (2))
From Fig. 5 we can mathematically express H ( z ) as
H ( z ) = g r ( z ) A W t b .Using Theorem 2 each of the components zPkFk( z ) / M in (5) can be implemented as in Fig.
5 . From ( 6 ) ,the level-vector for C k F k( z ) / M will be
bk
c
W-k',
k=O
and the numbers a/are related to the levels 0, by a/ =
f%)
.
as
q/ =
y/w-'L-'"/2
yI
sin (nLl/M)/sin ( n l / M ) ,
,
'
'
0Ir
(15)
g ' ( z ) = [Go(zM)z - ' G , ( t M )*
=
g%)C
(16a)
C = AR*A
(16b)
where A is the upper L x L submatrix of B , i.e.,
with
A
0 5 1 IM
-
1.
( 10)
Since L and M are relatively prime, one can verify that y I
# 0 for 0 II 5 M - 1. Pictorially, the theorem says
that the multilevel filter can be implemented as in Fig. 5
where g T ( z )is an M-input one-output system defined by
~ - ( ~ - lG )M
-
l<zM>]
(11)
- - 7--
Wk'L-l)Uk,L-I 0
where C A A W fB is M x L. In view of the zero-valued
entries in (15), we can rewrite
Remark: Notice that the quantity q1 can be simplified
1
* * *
O s k s L - 1
(9)
0 5 I 5 M - 1
C E i ' p;w-".
.
WkUk,I
= [t!dk.O
which takes the place of b in Fig. 5 . Defining the M X L
matrix B = [ b obl . . bL- I ] we can express f ( z ) as
L- I
.
(14)
H(z)
(8)
where
q1 A
(13)
and substituting into P ( z ) , we obtain P ( z ) = C E i '
qlz-'B, (zM) so that
M- I
H(z)
c z-'B,(zM)
/=0
=
[Douo Diu1
DL-IuL-I]'.
(17)
In terms of these notations, we can implement the synthesis bank { Fk ( z ) } as in Fig. 6. Note that the advance
operators z h can be avoided in practice by inserting an
. front of all the L inputs to C.
overall delay z - ( L - I ) in
Summarizing, we have proved:
Theorem 3: Let x ( n ) in Fig. 1 be a low-pass signal
band limited to I w I c L n / M . Then x ( n ) can be recovered from the nonuniformly sampled version (i.e.,
from the set of samplesx(nM - k ) , 0 5 k 5 L - 1 ) as
follows: design a low-pass filter P ( z ) with response as in
Fig. 3(a). With PI ( z ) denoting the type 1 polyphase com~~
-
1930
IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. A N D SIGNAL PROCESSING. VOL. 38. NO. I I . NOVEMBER 1990
The synlhesis bank (Fg (4F,(zl
...
F L -,IO)
Fig. 6. The complete multilevel synthesis bank, implemented in terms of
G,(z)andC.
ponents of this filter, define G, ( z ) as in (12). Compute
the matrix C from (16b). Then R ( n ) = x ( n ) (i.e., we
have perfect recovery from the nonuniformly decimated
signal) iff (z) defined in ( 5 ) satisfies (16a).
In practice one can replace P ( z ) with an FIR filter and
compute the FIR components G, ( z ) in Fig. 6 using (12).
The quantities A, R, A are not affected by this choice.
The cost of the reconstruction system is equal to that of
implementing P ( z ) and the matrix C. Some useful features of the matrix C are stated next.
Lemma I: The M X L matrix C has the following properties: 1)All the entries of the 0th row are equal to unity;
2 ) Cik = 0, f o r i Ik S L - 1, 1 Ii IL - 1 ; 3) c,k
= 0, f o r 0 Ik 5 i - M
L - l,M - L + 1 Ii I
M - 1; and 4)all entries are real valued.
To physically visualize the meaning of this, we display
C f o r M = 7, L = 4 below:
+
c=
/xxool
Iooxxl
x x x o
o x x x
L O O O X J
where x denotes arbitrary real numbers. There are two
"triangles" of zeros in this matrix. These are induced by
properties 2 and 3 of the Lemma. Because of Property 4,
the approximations Fk (z) have real-valued impulse responses (as long as the approximation P ( z ) has real coefficients). So f ( n) is real for real x ( n ) , a desirable result.
According to the lemma there are L2 trivial entries (0's
and 1's) so that at most ML - L2 real multipliers are
involved in implementing C. A proof of the Lemma is
included in the Appendix.
C. Practical Implementation of the Reconstruction
Scheme
If P ( z ) has the ideal response of Fig. 3(a) then aliasing
is completely eliminated (i.e., A / ( z ) = 0 , 1 # 0 ) and the
distortion function T ( z )defined in (4) is T ( z ) = 1 so that
R(n) = x ( n ) . We now consider the effects of replacing
P ( z ) with a nonideal filter. Since the ideal P ( z ) has real
coefficients, we restrict the approximant to have real coefficients. Evidently, aliasing is only approximately cancelled, and t( n ) is only appr9ximately band limited. Apart
from these obvious effects, X ( z ) differs from X ( z ) for another reason, viz., the distortion function T ( z ) is not
unity. Thus, even after the aliasing terms have been satisfactorily reduced, T ( z ) introduces amplitude and phase
distortions. The following important result tells us what
the value of T ( z) will actually be.
Theorem 4: Consider the system of Fig. 1 where the
synthesis bank is implemented as in Fig. 6, with GI ( z )
as in (12)where PI (z) are the type 1 polyphase components of an arbitrary transfer function P ( z ) , and C is as
in (16b). Then the quantity T ( z ) defined in (4) is given
by T ( z ) = P ( z ) .
A proof is included in the Appendix. The theorem holds
no matter what P ( z ) is, and indicates how phase distortion can be eliminated: just choose P ( z ) to have linear
phase! For notational convenience we take P ( z ) to have
zero phase so that the (real-valued) impulse response p ( n )
is symmetric with respect to n = 0. The passband ripples
of P ( z ) directly govern the extent of amplitude distortion
of the reconstruction system. It is for this reason that we
prefer to design P ( z ) (rather that the bandpass response
B ( z ) ) to be an optimal response (say, in the equiripple
sense) so that the amplitude distortion 1 T ( e '") 1 is minimized in the same sense.
1 ) Imposing "Mth Band '' Property on P ( z ) : Before
proceeding further, another important constraint should
be imposed on P ( z ) . The key observation behind the derivtion of the structure of Fig. 6 was the fact that the
shifted versions B ( e '"Wk ) of the ideal response B ( e J " )
can be used to construct a multilevel response of the form
in Fig. 4. In practice, however, we are less fortunately
situated. The function B ( e '") (which is uniquely determined by P ( z ) through (14))has a finite transition bandwidth, and nonzero passband and stopband ripples. The
ripples eventually affect the accuracy of the multilevel approximations, but the transition bandwidth can have a
more serious effect. This is demonstrated in Fig. 7. If the
overlaps of B ( e '") and B ( e '"W) are not carefully chosen, the sum B ( e J " ) B ( e J " W )can exhibit "bumps"
or "dips." As observed in [12],this effect can be overcome by forcing B ( z ) to be an Mth band filter.
Recall [13] that a zero-phase transfer function B ( z ) =
E, b ( n ) z - n is said to have the Mth band property if
b ( M n ) = 0 for all n # 0. This is equivalent to saying
that the type 1 polyphase component B&) is a constant.
We shall take this constant to be l / M for convenience.
Thus the Mth band property is equivalent to the important
property
+
M-l
c
k=O
B ( z W k ) = 1.
+
Now consider again the sum B ( e J " ) B ( e J " W )demonstrated in Fig. 7. If B ( e ' " ) has sufficiently high stopband attenuation, then the shifted responses B ( e'" W k), k
1931
VAIDYANATHAN A N D LIU: EFFICIENT RECONSTRUCTION OF BAND-LIMITED SEQUENCES
Fig. 7. The sum of B ( e '") and B ( e W ) .Unless the overlap is right. the
result can exhibit (a) bumps or (b) dips.
> 1 have negligible value in the region of overlap of
B(e'") and B ( e ' " W ) so that
M- I
C
k=O
B(ej'wk) = B(ej")
+ B(ej"W).
(20)
If B ( z ) has Mth band property, then the left-hand side of
(20) reduces to 1 so that the right-hand side of (20) approximates unity throughout the region of overlap of the
two terms. Thus no significant bumps or dips are exhibited. Next consider the summation Er=-: Pk B ( e'" W k)
which is used to obtain a multilevel filter. Once again in
the region where B ( e'") overlaps with B ( e'" W ), this can
be approximated by
I
=
Po[B(ej") + ~ ( e j " ~+) (]p1 - Po)B(ej"W)
= Po + ( P I - Po)B(e'"W)
(21)
which says that as long as B ( e J w has
) a monotone transition band, the change from the level Po to PI is monotone or "bump free."
It is thus desirable to take B ( z )to be an Mth band filter.
But in our design approach we do not have direct control
over B ( z ) because the prototype filter which we actually
design is the low-pass filter P ( z ) . However, since the
polyphase components of B ( z ) and P ( z ) are related as in
(14), it is clear that Bo(z)is constant if and only if P o ( z )
is constant. In other words, if P ( z ) is constrained to be
an Mth band filter, B ( z ) inherits this property.
2) The Design Procedure: Given the quantities M and
L , the first step is to design a zero-phase FIR Mth band
low-pass filter P ( z ) . Several techniques for FIR Mth band
filter design are available [ 131, [ 141. Standard specifications such as passband ripples, stopband attenuation and
transition bandwidth are chosen based on the extent of
amplitude distortion I T ( e'" ) I one is willing to tolerate in
the reconstructed signal. Once P ( z ) is designed the components PI ( z ) are known, and GI( z ) obtained from (12).
All quantities required to implement the synthesis bank of
Fig. 6 are now known.
3) Advantages of the New Method Over [5]: The structure of Fig. 6 works for arbitrary M , L, whereas those in
[ 5 ] were generated for special cases, without a unifying
polyphase framework. The new method also offers direct
control over the amplitude distortion I T(e'") 1 as explained in the paragraph following Theorem 4. The
method also has substantial computational advantages
over the earlier one, as we demonstrate in Section 111.
Another advantage follows from an easy result induced
by the Mth band property of P ( z ) , stated next.
Lemma 2: The Mth band property of P ( z ) (i.e., the
property " P o ( z ) = constant") implies that G o ( z )in Fig.
6 is a constant equal to 1/ M .
The proof is immediate from (12). By combining Lemmas 1 and 2 the reader can verify that the set of known
samples, viz., x ( n M - k ) , 0 Ik IL - 1 are not recomputed by the structure of Fig. 6; they are simply
passed through the synthesis bank, so their values are not
affected by rounding errors of any sort (or by the design
of P ( z ) ) . Only the subset of missing samples are computed by the synthesis bank.
4) A Disadvantage of the New Method: The method
does not offer direct control over the attenuation provided
by the alias-component transfer functions A/ ( z ) , 1 > 0 ,
even though this attenuation improves as P ( z ) moves
closer to ideal. The quantity I A/ (e'") 1 is required to be
small in the region where I X(e'"W-') 1 can have significant energy, but it is not clear how a prescribed specification on this attenuation can be satisfied optimally during
the design of the prototype P ( z ) . A lucky exception to
this is the L = 2, M = 3 case.
111. THEL = 2, M = 3 CASEA N D A DESIGN
EXAMPLE
We shall now demonstrate the ideas of the previous section by explicitly calculating the various matrices and
functions for the special case where L = 2 and M = 3. In
this case the vectorfT(z) is given by
fYz)
=
f
=
[ G o ( z 3 )z - I G I ( z 3 )z - ~ G ~ ( z ~ ) ] (22)
C.
z-lFl(z)]
To find C, notice from the definitions of A, R, and A that
where W = e -J2*I3.
Explicit computation shows
c=
[: :]
0 -1
which is real and indeed has the properties advertised in
Lemma 1 . Substituting into (22) we arrive at
F o ( z ) = 3 [ G o ( z 3 )+ z - ' G I ( z 3 ) ]
C 1 F 1 ( z )= ~ [ G o ( z '-) z - ~ G ~ ( z ~ ) ](24)
.
From (10) we verify that yo = 2, y I = I , and y2 = -1,
so from (12) G o ( z ) = P0(z)/2,G l ( z ) = P l ( z ) and
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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH. A N D SIGNAL PROCESSING. VOL 38. NO. I I . NOVEMBER 1990
G 2 ( z ) = - P 2 ( z ) . Next we evaluate4 ( z ) from (3) using
(24). These turn out to be
b
A
2x
0
3
Notice that by definition, A o ( z ) is the distortion function
T ( z ) . We know from Theorem 4 that this is supposed to
equal P ( z ) . This indeed is the case as can be verified by
replacing G, ( z ) in (25a) in terms of PI (z).
Imposing the third band constraint on P ( z ) results in
P , ( z ) = 2/3 so that P ( z ) =
z - ' P , ( z ) z-,P,(z).
Assuming further that P ( z ) is a real-coefficient zero-phase
filter, we have P ( z ) = P ( z - ' ) . From this it follows that
P2(z) = ZPI( z - I ) . These relations among PI ( 2 ) induce
relations among GI ( z ) ' s in view of (12). Summarizing,
we have
4+
+
3
8
,
x
0
4 2
2%
3
W
d
Fig. 8. Qualitative plots of I A, ( e ' " ) 1 for L = 2, M = 3 case. (a) I = 0,
(b) I = 1 , (c) I = 2.
the support of X( e'") coincides precisely with the passband of P ( z ) and the supports of X ( z W - ' ) and X ( z W - , )
coincide exactly with the stopbands of A I ( z ) and A2 (z).
Figs. 9(a) and (b) show the plots of I P(e'") I and
I A, ( e j " ) I. Usin4 the same test signal x ( n ) as in [5], we
obtain the plot I X ( e'") I shown in Fig. 9(c) for the reconstructed signal i ( n ) . This plot agrees well with the plot
of IX(e'") 1 shown in [ 5 ] . Since P ( z ) has linear phase,
f ( n ) is automatically free from phase distortion.
In order to compare this method with the one in [5], we
By using these in (25) and simplifying, we obtain the re- have to design the low-pass filter GL ( z ) and the Hilbert
lations
transformer GH( z ) in [ 5 , fig. 171 such that T ( z ) , A , ( z ) ,
A , ( Z ) = W [ A o ( z W )- 11
and A 2 ( z ) meet the same specifications as above. The
specifications of importance are: 1) passband width and
ripple of T ( z ) , 2) stopband regions and stopband attenso that 1 A , (e'") I = I A2(e-'") 1. Using the fact that uations of A I (z ) and A2(z). The reader can verify that
A o ( z W ) has zero phase, we give qualitative sketches of T ( z ) and A l ( z ) are related to GL ( z ) and G H ( z )as folI AI (e''") I in Fig. 8 for 1 = 0, 1 , 2. It is clear in this case lows: T ( z ) = C I G L( z ) , A , ( z ) = [ z - ' G L(z)/41 [ 1 that the attenuations provided by A I ( z ) and A, ( z ) are di- j h - ( h+ j ) G H ( z ) ]AlsoAP(z)satisfies
.
IA,(e'")I
rectly related to the passband ripple of A,( z ) (i.e., of the = IA,(e-"") I. Our aim now is to find the specifications
prototype P ( z ) ) . So the passband ripple of P ( z ) can be on GL ( z ) and G H ( z )so that T ( z ) ,A l ( z ) and A 2 ( z ) will
used to control the aliasing as well as amplitude distortion have same performance as in the new method. Let 4,,, &,
effects. Unlike in the arbitrary M , L case, we can there- E,,, and cS denote the passband edge, stopband edge, peak
fore control both aliasing and amplitude distortion di- passband ripple, and peak stopband ripple of GL ( z ) . Let
rectly by design of P ( z ) .
0, and Os be the band edges of G H( z ) (with same meaning
as in [5, fig. 18(b)]) and let 6 denote its peak passband
A . A Design Example
ripple. By sketching the functions I T( e J " ) 1, 1 A , (e'") 1,
We shall assume that x ( n ) is band limited to I w I IU,, and ] A 2 (e'") 1, the reader can verify that this method
where up < 2a/3. The gap A w L! 2a/3 - U,, helps us meets the same requirements as in the previous paragraph
to use filters with nonzero transition bands. Since the dis- if we choose
= 2n/3 - A w , & = 2n/3
A m , cp =
tortion function T ( z ) = P ( z ) , we shall choose P ( z ) to eS = 0.001, 0, = A w , and 6 = 0.002. We choose Os =
be low pass with passband edge at U,,. Our specifications a - 0, for reason described in [5]. These specifications
for P ( z ) in this example will be I ) w p = 2a/3 - 0 . 0 3 4 ~ can be met by a linear phase FIR GL( z ) of order 96, and
so that A w = 0.034a, and 2) peak passband ripple 0.001. a linear phase Hilbert transformer G H(z) or order 98. The
Because of the third band constraint on P ( z ) , its stopband number of multipliers to implement these are: 49 for
features cannot be independently specified [ 131. Several G L ( z ) and only 25 for G H ( z ) (due to symmetry of
standard methods exist for design of third band filters [ 131, I G H(e'") I with respect to a / 2 [15]). So this requires a
[14]; here we use the method in [17], which is a Remez- total of 76 multipliers (counting s and 1 - c in [5, fig.
type exchange algorithm, providing equiripple passband. 171).
The order of P ( z ) for the above specifications turns out
Finally, we shall compare these methods with the more
to be 94. From (27), the stopband attenuations provided traditional way [6] to fractionally compress a signal x ( n )
by A , ( z ) and A2 ( z ) are 60 dB. The reader can verify that band limited as above. This is shown in Fig. 10. Here
+,,
-
+
.
I
I933
VAIDYANATHAN A N D LIU: EFFICIENT RECONSTRUCTION OF BAND-LIMITED SEQUENCES
X(n) 3f+I-
M ( z ) m V ( n )
Fig. 10. The most well-known method for fractional decimation and reconstruction.
Method 111: the one reported in this paper.
It remains to estimate the cost of method I now. By
sketching the various internal waveforms in Fig. 10 one
can see that L ( z ) can be designed to be a low-pass halfband filter [13] with a large transition bandwidth ( = n / 3
Au). So its cost is very low, and we can afford to make
its in-band ripples sufficiently small so that the accuracy
of the reconstructed signal v ( n ) is determined primarily
by M ( z ) . In order to meet the same requirements on this
accuracy as methods I1 and 111, the following specifications were used for L ( z ) and M ( z ) . 1) L (z): half-band
linear-phase FIR with peak passband and stopband ripples
= 0.0001, and transition bandwidth = n / 3 + A w . 2)
M ( z): third-band linear-phase FIR with peak passband
and stopband ripples = 0.001 and transition bandwidth =
A u . The orders of these filters turn out to be 26 and 190,
respectively. By exploiting the half-band and third band
properties of L ( z ) and M ( z), respectively, (and also their
linear-phase property) we can implement these with 7 and
64 multipliers, respectively, i.e., a total of 71 multipliers.
The new method (method 111) requires fewer than half as
many multipliers as either of the earlier methods.
+
-100.000
I
0.
0.200
0.400
8.600
0.800
I
1.000
NORIIALIZEO FREQUENCY
(a)
-20.000
E
-40.000
I
1
w
0
a
-60.000
.
-80.000
-
z
Is
a
E
-100.000
'
0.
'
I
.'
0.200
0.400
I
0.600
0.800
1.1100
NORHALIZED FREQUENCY
(b)
1.000
0.800
0
LL
z
v)
0.600
0
LL
0.400
3
I
c
LI)
z
g
0.200
0,
0.
0.1L70
0.200
0.300
0.400
0.5011
B. Comparison Based on Multiplications Per Unit Time
It is often more appropriate to perform the complexitycomparison based on the number of multiplications per
unit time (defined as the interval between successive samples of the original signal x ( n ) ) . For method I, the filters
L ( z ) and M ( z ) can be implemented very efficiently by
simultaneous exploitation of the presence of decimators
and interpolators as described in [12], [ 161. If this is done,
one can verify that L ( z ) requires only 3 MPU's and M ( z )
requires 22 so the total is 25 MPU's. For method 11, it is
not possible to rearrange the flowgraph of [ 5 , fig. 171 in
order to exploit the presence of decimators and interpolators so that the number of MPU's remains equal to 76.
For method I11 we can implement the reconstruction system as in Fig. 11, requiring a total of only 11 MPU's.
Table I summarizes the comparison of the three methods.
The advantage offered by method I11 is obvious.
NORHRLIZEO FREQUENCY
(C)
Fig. 9. Plots of magn-itudes of (a) P ( e ' " ) = T ( e ' " ) . (b) A?( e'"), and (c)
X ( e ' " ) . for the design example.
y ( n ) is the compressed signal and v ( n ) is the reconstructed signal at the original rate (i.e., the rate of x( n ) ) .
To facilitate our discussions, we shall assign the following names to the three methods:
Method I: the one shown in Fig. 10 (earliest method
161).
Method 11: the one reported in [ 5 , fig. 171 (recent
method).
IV. NONCONSECUTIVE
SAMPLING
A N D MULTIBAND
SIGNALS
In Section I1 we retained L consecutive samples x ( n M
- k ) , 0 I k I L - 1 from each length-M segment of
x ( n ) . We now consider two generalizations: first, we ret a i n x ( n M - n k ) , O s k s L - lwherenkarenotnecessarily consecutive, but satisfy 0 5 no < n l < * * * <
nL- < M. Second, we do not restrict x (n ) to be low pass
but consider it to be a multiband signal. Such a signal is
one for which X ( e J w) is nonzero in a subset of L regions
out of the M regions shown in Fig. 12. Let RIq,0 Iq 5
,
1934
IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. A N D SIGNAL PROCESSING. VOL. 38. NO. 1 1 . NOVEMBER 1990
W M )
3GlW
X(nM-1)
3G2W
Fig. 13. Prototype zero-phase filter for multiband case.
2-1
-1
Fig. 1 1 . Rearranging the reconstruction system for the design example
(method 111).
TABLE I
COMPARISON OF THE THREE
RECONSTRUCTION
METHODS
IN TERMS
OF
NUMBER
OF MULTIPLICATIONS
Region M -1
Region 0 Region 1
/ I d 1
-.a
I
d
{
2x
Fig. 12. Definition of the M regions for the multiband case.
L - 1 denote these L regions, let C be the set { lo, lI,
* * , l L - I } , and let 63 be the union of these L regions
(i.e., 63 is the support of X ( e ' " ) ) . Since M and 6: completely characterize the frequency bands, we shall say that
x( n ) is an (M, 6: )-BL signal (extension of the a-BL jargon! ). Since the total spectral occupancy of x ( n ) is only
27rL/M, it is possible to compress the signal by M I L .
Suppose we compress x ( n ) by nonuniform nonconsecutive decimation, i.e., by retaining x ( n M - n k ) , 0 Ink
IL - 1. Can we still recover x ( n ) by use of a synthesis
bank { F k ( Z ) } ?
The answer is yes, provided the set { n k } is chosen
properly. In particular, if { nk } are consecutive, such reconstruction is always possible. We shall now quantitatively state these and related results. The developments
are analogous to those in Section I1 though notationally
more tedious. Proofs can be found in [ 171.
The nonconsecutive nonuniform decimation can be represented as in Fig. 1 except that the delay elements are
now z -"',0 s k 5 L - 1. Once again 2(n) suffers from
the same set of distortions as in Section 11.
=
(1/M)
[Z-"~F,(Z)
Define f'(z)
z - ' ~ .IFL- l ( z ) ]T, V to be the L X L matrix with elements
VI = Wflh/m, and e4 to be the qth unit vector. If { Fk ( z ) }
mk
is the set of L synthesis filters resulting in perfect reconstruction (i.e., i ( n ) = x ( n ) ) , then V E ; I f ' ( e ' " ) = e,
in region Rly, where Eq = diag [ W'yfl0, * , W'@-I1 . so
-
I
T
1
we can solve forf' ( e'") provided that V is nonsingular.
This, in general, is not guaranteed for arbitrary { nk } and
{ l , , , } . In the special case where nk are consecutive integers or when 1, are consecutive integers, V' does become
nonsingular because it (or its transpose) is then a Vandermonde matrix with distinct columns. Assuming that V' is
nonsingular, the solution vector f' is such that each
z V f l h F k ( zis) again a multilevel filter, with Fk(e'") = 0
for w 63. So we can again obtain all these ideal filters
in terms of a prototype zero-phase Mth band filter with
response B' ( e'") shown in Fig. 13. Equivalently, we can
use the polyphase components of a zero-phase Mth band
prototype P' ( e'") whose response equals unity for w E 63
and zero otherwise. The advantage of using P ' ( e J " )is
that, when the ideal filter is replaced with a practical approximation, the distortion function T( z) is equal to P' (z)
so that we have direct control over the specifications of
T ( z )while designing P ' ( z ) . The resulting synthesis bank
can once again be expressed as in Fig. 6 with some obvious modification such as using primed versions, etc. The
matrix C is also appropriately replaced. This matrix C
completely summarizes the nonuniform decimation pattern { nk } and the multiband pattern { 1, }. The expression
for f' ( z ) in terms of Pi ( z ) (which are the polyphase
components of P ' ( z ) ) contains terms of the form
PA(z)/qA where q: =
W-"'I. Unlike q1 defined in
(lo), q,', can be zero, so the use of P ' ( z ) as a prototype
may not be feasible. However, it is always possible to use
B ' ( z ) in order to generate all the L multilevel filters
z-"lFk(z). Once again, if I,,,, 0 Im 5 L - 1, are consecutive then q; are guaranteed to be nonzero.
The usefulness of these results on generalized nonuniform decimation of multiband signals will now be demonstrated by a simple example. Let L = 2 , M = 4, and
let X(e'") be as in Fig. 14(a). Here lo = 1, l1 = 3. Our
aim is to keep two out of every four samples from x ( n )
in such a way that x ( n ) can eventually be reconstructed.
If we choose no = 0, n 1 = 2 (uniform decimation by two)
then V' = [ ; $1, which is singular. So the reconstruction
scheme will fail. On the other hand, if we choose no = 0
and n l = 1 (nonuniform decimation) then V =
which is nonsingular so that the reconstruction can be carried out. Note, incidentally, that unlike in Section I1 (lowpass case), it makes sense to have nontrivial common factors between L and M here. The reason why uniform decimation fails in this example can also be seen in the frequency domain: When x ( n ) is decimated uniformly by
two, the result has Fourier transform [X(e""12) +
X ( e J ' " - 2 " ' / 2 ] / 2and the two components overlap (Fig.
14(b)). So x ( n ) cannot be recovered.
[I El]
I935
V A I D Y A N A T H A N A N D LIU: EFFICIENT RECONSTRIK’TION O F BAND-LIMI T E D SEQUENCES
Fig. 14. Example of (a) a multiband X ( e J u ) and (b) the uniformly decimated version.
V. MULTIDIMENSIONAL
EXTENSIONS
The results of the previous sections easily extend to the
multidimensional case. We shall briefly summarize the
results here using, for convenience, the 2D case. Some
details can be found in [ 171. The extensions are based on
integer sublattices and cosets [ 181-[20] which are related
to the notions of generalized (non rectangular) decimators
and generalized polyphase components [20].
A. Review from Integer Lattice Theory
The set Z 2 of all 2 x 1 integer vectors is called the 2D
integer lattice. Any 2 X 2 integer matrix D with det [D]
= M > 0 generates a sublattice ADof Z 2 , which is the
set of all integer vectors of the form Dn,n E 2 * .A coset
AD(n,)of AD, generated by any integer vector n, is the
set of all’integer vectors of the form n, Dn,n E 2 ’ .
The vector n, is also loosely called a “shift-vector.’’ An
important result in lattice theory says that for any integer
matrix D with determinant M > 0, there exists a set of M
shift vectors { no, n l , * * , n M - I } such that any n E Z 2
can be expressed as n = n, + D m for some m E Z 2 . Thus
Z 2 is a union of M cosets. Note the analogy to the 1D
case: any integer n E 2 can be represented as n, + M m ,
0 5 n, 5 M - 1, m E 2 . This analogy can be used to
define polyphase decomposition for multidimensional signals.
~ . 6 denote the set of vectors
Now let E 2 ~ T D - Let
of the form Eu, U E [ 0, 1 )*. Then there exists a set of M
, m M - } such that any 2D
integer vectors { mo,m l ,
frequency w E [0, 2 ~ can
) be~ expressed as w = wo +
E m ,for some wo E 6 ,and some m,in the above set. For
a fixed m, denote R,, to be the set of all w that can be
expressed as wo + Em,,w,, E 6.Then [0, 2 ~ is )the~
union of the M regions R,”<.
+
-
B. Application to Nonuniform Decimation
Any integer matrix D with determinant M > 0 can be
used to partition the integer set Z 2 into M cosets determined by the M integers n,.From D we can find a corresponding partitioning of the frequency domain [0, 2 ~
into M regions, determined by the M integer vectors m,.
We say that the 2D signal x ( n ) is multiband bandlimited
if X ( w ) is nonzero only in L out of these M regions, L <
M . Let the L regions be R , , , 0 Ii IL - 1. We should
now appropriately define a nonuniform decimation scheme
which keeps L out of M samples. Suppose we retain the
samples x ( Dn + n,)with 0 Ii IL - 1. In other words,
only the samples belonging to the first L of the M cosets
are retained. Is it possible to reconstruct x ( n ) from these
samples by using a 2D version of the synthesis bank? The
answer again is in the affirmative, provided the coset of
retained samples is carefully chosen (elaborated below).
The decimation and reconstruction process can be formulated in a way analogous to that in Fig. 1. Thus, we
can use the system shown in [20, fig. 13, with a subset of
analysis filters taken to be 2D advance operators and the
others to be zero. The corresponding set of L synthesis
filters which gives rise to perfect recovery can again be
obtained by solving a set of L equations. There are L sets
of such equations, one for each of the L regions R,,. All
these sets of equations involve the inversion of a,fixed L
x L matrix, whose entries are given by e-12r”lJD”‘, 0 I
i , k IL - 1. So the reconstruction can be carried out if
this matrix is nonsingular. For the special case where the
shift vectors nk take the form nk = km,0 Ik IM - 1,
m # 0,this matrix is guaranteed to be nonsingular prokm) for L convided the retained samples are x ( D n
secutive values of k. This is true regardless of the subset
m,which represents the L regions of support of X( w ) .
+
APPENDIX
Proof of Lemma I : We shall first prove the fourth
property, i.e., that C defined as in (16b) is real. This result follows from an important property satisfied by the
columns U , of the matrix U 6 V - ’ , viz.,
D,u,= D ~ - l - , u ~ - l0 -5,i ,5 L
-
1.
(A.l)
To prove this property of u;s, note that Vui = e, where e,
is the ith unit vector. So JVu, = e,- I -, where J is the
flipping matrix defined in Section I. But from the definition of V we can verify JV = V * D L - I so that “JVu, =
eL- I - ,” implies “ V * DL- U , = e, - I - ,” which implies
“VD,*_,u: =
But we also have
e L - I -, which proves that U,- I -, = DF- I u f implying
(A. 1) indeed. Now A is given by (17) so that
JA =A*
(‘4.2)
in view of (A. 1). From the definition of R it also follows
that
RJ = A2R*.
(‘4.3)
By using (A.2) and (A.3) we obtain RA* = R J J A * =
A*R*A, which proves that A*RA* = A R * A , i.e., that C
is real. Now consider proving Properties 1-3. This is
equivalent to proving that R*A has these same properties.
If we use the following facts: 1) V is the upper L x L
)submatrix
~
of the M X L matrix R , 2) V is the inverse of
U, and 3) the rows of A in (17) are determined by the
columns U , of U, we can easily verify that R*A does satisfy these properties. This completes the proof of Lemma
1.
Proof of Theorem 4: Recall that T ( z ) is defined as in
(4),so T ( z ) = f T ( z ) l where 1 [ 11 . . 1 I T . From
1936
IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. VOL. 38. NO.
(16a), it thus follows that T ( z ) = g T ( ~ ) A R * A 1From
.
the definition of A and from the fact that UV = VU = I ,
we have A 1 = 1 so that R*A 1 = R * l . But from (10) we
have R*l = [ q o ql
* qu- I ]‘. Using this with (10) we
T
get T ( z ) = g T ( z ) u where u = [ y o y I
YM-11 .
Using (12) this gives T ( z ) = P ( z ) .
-
-
11,
NOVEMBER 1990
[18] E. Dubois, “The sampling and reconstruction of time-varying imagery with application in video systems,” Proc. IEEE, vol. 73, pp.
502-522, Apr. 1985.
[I91 D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1984.
[20] E. Viscito and J. Allebach, “Design of perfect reconstruction multidimensional filter banks using cascaded Smith form matrices,” in
Proc. IEEE Int. Symp. Circuits Sysr. (Espoo, Finland), June 1988,
pp. 831-834
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[I] C. E. Shannon, “Communications in the presence of noise,” Proc.
IRE, vol. 37, pp. 10-21, Jan. 1949.
[2] A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst., pp. 652-654, Nov. 1977.
[3] J. L. Brown, Jr., “Multichannel sampling of low-pass signals,” IEEE
Trans. Circuits Sysr., vol. 28, pp. 101-106, Feb. 1981.
[4] A. J . J e m , “The Shannon sampling theorem-its various extensions
and applications: A tutorial review,” Proc. IEEE, pp. 1565-1596,
Nov. 1977.
[5] P. P. Vaidyanathan and V. C. Liu, “Classical sampling theorems in
the context of multirate and polyphase digital filter bank structures,”
IEEE Trans. Acousr., Speech, Signal Processing, vol. 36, pp. 14801495, Sept. 1988.
[6] R. E. Crochiere and L. R. Rabiner, Mulrirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983.
[7] M. J . T . Smith and T . P. Barnwell, 111, “A new filter-bank theory
for time-frequency representation,” IEEE Trans. Acoust. , Speech,
Signal Processing, vol. 35, pp. 314-327, Mar. 1987.
[8] P. P. Vaidyanathan, “Theory and design of M-channel maximally
decimated quadrature mirror filters with arbitrary M, having perfect
reconstruction property,” IEEE Trans. Acousr., Speech, Signal Processing, vol. 35, pp. 476-492, Apr. 1987.
[9] M. Vetterli, “A theory of multirate filter banks,” IEEE Trans.
Acoust., Speech, Signal Processing, vol. 35, pp. 356-372, Mar. 1987.
1101 P. P. Vaidyanathan, “Quadrature mirror filter banks, M-band extensions and perfect-reconstruction techniques, ” IEEE ASSP Mag. , vol.
4, pp. 4-20, July 1987.
[ l l ] M. Bellanger, G. Bonnerot, and M . Coudreuse, “Digital filtering by
polyphase network: Application to sample rate alteration and filter
banks,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 24,
pp. 109-1 14, Apr. 1976.
1121 P. P. Vaidyanathan, “Multirate digital filters, filter banks, polyphase
networks, and applications: A tutorial,” Proc. IEEE, to be published.
1131 F. Mintzer, “On half-band, third-band, and Nth band FIR filters and
their design,” IEEE Trans. Acousr., Speech, Signal Processing, vol.
30, pp. 734-738, Oct. 1982.
[I41 P. P. Vaidyanathan and T . Q. Nguyen, “Eigenfilters: A new approach to least squares FIR filter design and applications including
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[I51 L. R. Rabiner and B. Gold, Theory and Application of Digital Signal
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[I61 C.-C. Hsiao, “Polyphase filter matrix for rational sampling rate conversions,” in Proc. IEEE Inr. ConJ ASSP (Dallas, TX), Apr. 1987,
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1171 V. C. Liu, “One- and two-dimensional digital multirate systems with
applications in subsampling and band-limited signal reconstruction,”
doctoral dissertation, Dep. Elec. Eng., California Institute Technology, Pasadena, CA, 1989.
P. P. Vaidyanathan (S’80-M’83-SM’88) was
born in Calcutta, India, on October 16, 1954. He
received the B.Sc. (Hons.) degree in physics, and
the B.Tech. and M.Tech. degrees in radiophysics
and electronics from the University of Calcutta,
India, in 1974, 1977, and 1979, respectively, and
the Ph.D. degree in electrical and computer engineering from the University of California, Santa
Barbara, in 1982.
He was a postdoctoral fellow at the University
of California, Santa Barbara, from September
1982 to February 1983. In March 1983, he joined the California Institute
of Technology, Pasadena, as an Assistant Professor of Electrical Engineering, and is currently an Associate Professor in the same Department.
His main research interests are in digital signal processing, multirate filter
bank systems, filter design, adaptive filtering, and multivariable system
theory.
Dr. Vaidyanathan served as the Vice-chairman of the Technical Program Committee for the 1983 IEEE International Symposium on Circuits
and Systems and as an Associate Editor for the IEEE TRANSACTIONS
ON
CIRCUITS
A N D SYSTEMS,
for the period 1985 to 1987. He received the award
for Excellence in Teaching from the California Institute of Technology, for
the year 1983-1984. He was also a recipient of NSF’s Presidential Young
Investigator Award, starting from 1986. In 1989 he received the IEEE ASSP
Senior Award for his paper on multirate perfect-reconstruction filter banks.
In 1990 he received the S. K. Mitra Memorial Award from the Institute of
Electronics and Telecommunications Engineers, India, for his paper (with
coauthor V. Liu) in the IETE Journal.
Vincent C. Liu (S’85-M’89) was born in Hong
Kong in 1961. He received the B.E. degree in
electrical engineering from the State University of
New York at Stony Brook in 1984, the M S. degree in electrical engineering from the California
Institute of Technology, Pasadena, in 1985, and
the Ph.D. degree in electrical engineering from
the California Institute of Technology in 1989.
He is now a Senior Engineer with the Videocipher Division of the General Instrument Corporation, San Diego, CA His research interests
include digital signal processing, two-dimensional systems, and multirate
filter bank applications.
Dr. Liu was a recipient of the Schlumberger Fellowship, and is a member of Tau Beta Pi.